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I have developed magnitude plots for basic one-pole and two-pole filters derived from various Physics PDF's and tutorials I have found online and calculated from:

https://www.desmos.com/calculator/mjudrnexbo

But I cannot figure out what the correct plots for any simple four-pole resonant LPF, HPF, BP, & BR filters would be in the same terms.

Any help on what equivalent simple four equations would be? Thanks.

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  • $\begingroup$ there are a lot more degrees of freedom for 4-pole filters than for 2-pole filters. $\endgroup$ – robert bristow-johnson Dec 17 '18 at 6:38
  • $\begingroup$ I'm looking for the most computationally simple 4-pole equations that will permit resonance for LPF/HPF and bandwidth for BP/BR to be controlled by a simple "Q" function. $\endgroup$ – Mike Dec 18 '18 at 4:45
  • $\begingroup$ Mike, each of the 2-pole filters that make up your 4-pole filter has their own independent Q. $\endgroup$ – robert bristow-johnson Dec 18 '18 at 5:31
  • $\begingroup$ do you wanna do the Moog 4-pole filter? $\endgroup$ – robert bristow-johnson Dec 18 '18 at 5:31
  • $\begingroup$ or do you want a 4-pole Butterworth or 4-pole Tchebyshev? $\endgroup$ – robert bristow-johnson Dec 18 '18 at 5:34
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There are several optimality criteria to choose from: Butterworth, Chebyshev (type $1$ and $2$), Cauer, etc., and each of them will give you a different magnitude response.

Butterworth filters have an especially simple magnitude (gain) function. For a low pass filter of order $n$ (i.e., with $n$ poles) you get

$$G_{LP}(f)=\frac{1}{\sqrt{1+\left(\frac{f}{f_c}\right)^{2n}}}\tag{1}$$

where $f_c$ is the $3$-dB cut-off frequency. The gain of an $n^{th}$ order Butterworth high pass filter is

$$G_{HP}(f)=\frac{\left|\frac{f}{f_c}\right|^{n}}{\sqrt{1+\left(\frac{f}{f_c}\right)^{2n}}}\tag{2}$$

The general formulas for band pass and band stop filters are more complicated. For a fourth order band pass filter you get

$$G_{BP}(f)=\frac{f_c^2f^2}{\sqrt{f^8-4f_0^2f^6+(f_c^4+6f_0^4)f^4-4f_0^6f^2+f_0^8}}\tag{3}$$

where $f_0$ is the center frequency:

$$f_0=\sqrt{f_{l}f_{u}}\tag{4}$$

with $f_l$ and $f_u$ the lower and upper cut-off frequencies, respectively. The frequency $f_c$ is given by

$$f_c=\frac{f_u^2-f_0^2}{f_u}\tag{5}$$

Finally, the magnitude of a fourth order Butterworth band stop filter is given by

$$G_{BS}(f)=\frac{(f_0^2-f^2)^2}{\sqrt{f^8-4f_0^2f^6+(f_c^4+6f_0^4)f^4-4f_0^6f^2+f_0^8}}\tag{6}$$

where the center frequency $f_0$ and $f_c$ are given by $(4)$ and $(5)$, respectively.

The figure below shows the magnitudes computed according to Eqs $(1)$, $(2)$, $(3)$, and $(6)$. The cut-off frequency for the low and high pass filters was chosen as $f_c=4$, and the lower and upper cut-off frequencies for the band pass and the band stop filters were chosen as $f_l=2$ and $f_u=4$, respectively, resulting in a center frequency $f_0=2\sqrt{2}$.

enter image description here

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  • $\begingroup$ Thank you very very much Matt. I need these equations to be resonant though with a Q value. Is there any chance you could provide the simplest equations that would allow a Q value? For bandpass/bandreject I'd ideally like the width of the band controlled by a simple Q value as well. Thanks so much if so. I do appreciate it. $\endgroup$ – Mike Dec 18 '18 at 4:42
  • $\begingroup$ Personally, I find it a bit simpler, in this case, to keep the lowpass prototype, and simply apply frequency transformations: $H(\frac x{f_p})$ for lowpass, $H(\frac{f_p}x)$ for HP, $H(\frac{x^2-f_c^2}{BW x})$ for BP, $H(\frac{BW x}{x^2-f_c^2})$ for BS. The only minor downgrade would be the need to precalculate $f_1$ and $f_2$ from solving the quadratics of BP and BS. $\endgroup$ – a concerned citizen Dec 18 '18 at 7:52
  • $\begingroup$ @Mike: Fourth order systems generally don't have a simple $Q$ factor (like second order systems). What you can do is simply concatenate (multiply) two second-order systems with the same $Q$ factor to obtain a (very specific) fourth-order system. $\endgroup$ – Matt L. Dec 18 '18 at 11:30
  • $\begingroup$ With LPF and HPF one could use this method ... dsprelated.com/showthread/comp.dsp/… $\endgroup$ – Juha P Dec 18 '18 at 15:21

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